It has been suggested that Homotopy lifting property be merged into this article or section. (Discuss)

In algebraic topology, a fibration is a continuous mapping Image File history File links Please see the file description page for further information. ... In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property is a technical condition on a continuous function from a topological space X to another one, Y. It is designed to support the picture of X above Y, by allowing a homotopy taking place in Y... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...

E → B

satisfying the homotopy lifting property with respect to any space. Fiber bundles constitute important examples; but in homotopy theory any mapping is 'as good as' a fibration - i.e. any map can be decomposed as a homotopy equivalence into a "mapping path space" followed by a fibration. In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property is a technical condition on a continuous function from a topological space X to another one, Y. It is designed to support the picture of X above Y, by allowing a homotopy taking place in Y... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ...

A fibration with the homotopy lifting property for CW complexes (or equivalently, just cubes Iⁿ) is called a Serre fibration, in honor of the part played by the concept in the thesis of Jean-Pierre Serre. This thesis firmly established in algebraic topology the use of spectral sequences, and clearly separated the notions of fiber bundles and fibrations from the notion of sheaf (both concepts together having been implicit in the pioneer treatment of Jean Leray). Because a sheaf (thought of as an étale space) can be considered a local homeomorphism, the notions were closely interlinked at the time. In topology, a CW complex is a type of topological space introduced by J.H.C. Whitehead to meet the needs of homotopy theory. ... Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In homological algebra, especially applied to algebraic topology or group cohomology, a spectral sequence is a sequence of differential modules (En,dn) such that En+1 = H(En) = ker dn / im dn is the homology of En. ... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... Jean Leray (7 November 1906-10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology. ... In topology, a local homeomorphism is a map from one topological space to another that respects locally the topological structure of the two spaces. ...

The fibers are by definition the subspaces of E that are the inverse images of points b of B. Fibrations do not necessarily have the local cartesian product structure that defines the more restricted fiber bundle case, but something weaker that still allows 'sideways' movement from fiber to fiber. One of the main desirable properties of the Serre spectral sequence is to account for the action of the fundamental group of the base B on the homology of the total space E. In mathematics, the Cartesian product (or direct product) X Y of two sets X and Y is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. This concept is named after Ren Descartes. ... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...

The projection map from a product space is very easily seen to be a fibration. Fiber bundles have 'local trivialization's — such cartesian product structures exist locally on B, and this is usually enough to show that a fiber bundle is a fibration. More precisely, if there are local trivializations over a "numerable open cover" of B , the bundle is a fibration. Any open cover of a paracompact space - for example any metric space, has a numerable refinement, so any bundle over such a space is a fibration. The local triviality also implies the existence of a well-defined fiber (up to homeomorphism), at least on each connected component of B. In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... In mathematics, something is said to occur locally in the category of topological spaces if it occurs on small enough open sets. ... In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement. ... In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc. ... Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ...

In the Category of small categories, a functor p : E → C from a category E to a category C is a fibration iff for every object X of E and every map γ into pX in C there exists a cartesian morphism into X over γ (see also semidirect product). In mathematics, specifically in category theory, the 2-category of small categories is the 2-category whose objects are small categories, whose arrows are functors and whose 2-arrows are natural transformations. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In mathematics, in particular category theory, given a functor p:E→C from a category E to a category C, a morphism f : X → Y in E is cartesian (with respect to p) when for each object Z of E and each morphism γ : pZ → pX in C, the function... In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ...

Fibrations in closed model categories

A fibration in a closed model category C is an element of the class of morphisms of C called the fibrations of C. These are formally dual to the cofibrations in the opposite category C^op and in particular they are closed under composition and pullbacks. Any morphism in such a category can (by definition) be factored into the composition of a acyclic cofibration followed by a fibration or a cofibration followed by a acyclic fibration, where the word "acyclic" indicates that the corresponding arrow is also a weak equivalence. (In the original treatment, due to Daniel Quillen, the word "trivial" was used instead of "acyclic.") The lifting property comes from one of the axioms for a model category which ties together fibrations and cofibrations by such lifts. In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms (arrows): weak equivalences, fibrations and cofibrations. These abstract from a conventional homotopy category, of topological spaces or of chain complexes (derived category theory). ... In mathematics, in particular homotopy theory, a continuous mapping i: A → X, where A and X are topological spaces, is a cofibration if it satisfies the homotopy extension property. ... In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in... In category theory, a branch of mathematics, the pullback (also called the fiber product) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. ... In mathematics, a weak equivalence is a notion from homotopy theory which identifies complexes that have the same basic shape in terms of their homology groups. ... Daniel Quillen (born June 27, 1940) is an American mathematician, a Fields Medallist, and the current Waynflete Professor of Pure Mathematics at Magdalen College, Oxford. ... In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ...